Proof of three conjectures on determinants related to quadratic residues

Darij Grinberg, Zhi-Wei Sun, Lilu Zhao

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\r…